The Yamabe problem in differential geometry concerns the existence of Riemannian metrics with constant scalar curvature

Scalar curvature

In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...

, and takes its name from the mathematician Hidehiko Yamabe

Hidehiko Yamabe

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. Although claimed to have a solution in 1960, a critical error
in his proof was discovered by . The combined work of Neil Trudinger

Neil Trudinger

Neil Sidney Trudinger is an Australian mathematician, known particularly for his work in the field of nonlinear elliptic partial differential equations....

, Thierry Aubin

Thierry Aubin

Thierry Aubin was a French mathematician at Centre de Mathématiques de Jussieu who worked on Riemannian geometryand non-linear partial differential equations...

, and Richard Schoen

Richard Schoen

Richard Melvin Schoen is an American mathematician. Born in Fort Recovery, Ohio, he received his PhD in 1977 from Stanford University where he is currently the Anne T. and Robert M. Bass Professor of Humanities and Sciences...

 provided a complete solution to the problem as of 1984.

The Yamabe problem is the following: given a smooth, compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

  of dimension with a Riemannian metric , does there exist a metric conformal

Conformal map

In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

 to for which the scalar curvature

Scalar curvature

In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...

 of is constant? In other words, does a smooth function exist on for which
the metric has constant scalar curvature? The answer is now known to be yes, and was proved using techniques from differential geometry, functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

 and partial differential equations.

The non-compact case

A closely related question is the so-called "non-compact Yamabe problem", which asks: on a smooth, complete Riemannian manifold
which is not compact, does there exist a conformal metric of constant scalar curvature that is also complete? The answer is no, due to counterexamples
given by